\(\int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx\) [52]

Optimal result

Integrand size = 20, antiderivative size = 242 \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^3}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{4 \left (a^2+b^2\right ) f^4} \]

[Out]

1/4*(d*x+c)^4/(a-I*b)/d-b*(d*x+c)^3*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))/(a^2+b^2)/f+3/2*I*b*d*(d*x+c)^2*pol
ylog(2,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))/(a^2+b^2)/f^2-3/2*b*d^2*(d*x+c)*polylog(3,(a+I*b)*exp(2*I*(f*x+e))/(a
-I*b))/(a^2+b^2)/f^3-3/4*I*b*d^3*polylog(4,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))/(a^2+b^2)/f^4

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3812, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 f^3 \left (a^2+b^2\right )}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 f^2 \left (a^2+b^2\right )}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f \left (a^2+b^2\right )}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{4 f^4 \left (a^2+b^2\right )}+\frac {(c+d x)^4}{4 d (a-i b)} \]

[In]

Int[(c + d*x)^3/(a + b*Cot[e + f*x]),x]

[Out]

(c + d*x)^4/(4*(a - I*b)*d) - (b*(c + d*x)^3*Log[1 - ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*
f) + (((3*I)/2)*b*d*(c + d*x)^2*PolyLog[2, ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f^2) - (3*
b*d^2*(c + d*x)*PolyLog[3, ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/(2*(a^2 + b^2)*f^3) - (((3*I)/4)*b*d^3*
PolyLog[4, ((a + I*b)*E^((2*I)*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f^4)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3812

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^
(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^Simp[2*I*(e + f*x), x]/((a + I
*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Integer
Q[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4}{4 (a-i b) d}+(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{(a-i b)^2+\left (-a^2-b^2\right ) e^{2 i (e+f x)}} \, dx \\ & = \frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {(3 b d) \int (c+d x)^2 \log \left (1+\frac {\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f} \\ & = \frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^2}-\frac {\left (3 i b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f^2} \\ & = \frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^3}+\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{2 \left (a^2+b^2\right ) f^3} \\ & = \frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^3}-\frac {\left (3 i b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {(a+i b) x}{a-i b}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 \left (a^2+b^2\right ) f^4} \\ & = \frac {(c+d x)^4}{4 (a-i b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^3}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{4 \left (a^2+b^2\right ) f^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.19 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.43 \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\frac {b \left (\frac {2 i (c+d x)^4}{(a+i b) d}-\frac {4 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (c+d x)^3 \log \left (1+\frac {(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right )}{\left (a^2+b^2\right ) f}+\frac {3 d \left (-i a \left (-1+e^{2 i e}\right )+b \left (1+e^{2 i e}\right )\right ) \left (2 f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )+d \left (-2 i f (c+d x) \operatorname {PolyLog}\left (3,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )-d \operatorname {PolyLog}\left (4,\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )\right )\right )}{\left (a^2+b^2\right ) f^4}\right )}{4 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right )}+\frac {x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \sin (e)}{4 (b \cos (e)+a \sin (e))} \]

[In]

Integrate[(c + d*x)^3/(a + b*Cot[e + f*x]),x]

[Out]

(b*(((2*I)*(c + d*x)^4)/((a + I*b)*d) - (4*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))*(c + d*x)^3*Log[1 +
(-a + I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f) + (3*d*((-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*
I)*e)))*(2*f^2*(c + d*x)^2*PolyLog[2, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] + d*((-2*I)*f*(c + d*x)*PolyL
og[3, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))] - d*PolyLog[4, (a - I*b)/((a + I*b)*E^((2*I)*(e + f*x)))])))/
((a^2 + b^2)*f^4)))/(4*(a*(-1 + E^((2*I)*e)) + I*b*(1 + E^((2*I)*e)))) + (x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 +
d^3*x^3)*Sin[e])/(4*(b*Cos[e] + a*Sin[e]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (215 ) = 430\).

Time = 0.43 (sec) , antiderivative size = 1402, normalized size of antiderivative = 5.79

method	result	size
risch	\(\text {Expression too large to display}\)	\(1402\)

[In]

int((d*x+c)^3/(a+b*cot(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/4*d^3/(a+I*b)*x^4+1/(a+I*b)*c^3*x+1/4/d/(a+I*b)*c^4+I/f^4/(b-I*a)*b*e^3*d^3/(I*b-a)*ln(exp(2*I*(f*x+e))*a+I*
b*exp(2*I*(f*x+e))-a+I*b)+6/f/(b-I*a)*b/(a-I*b)*c^2*d*e*x+I/f^4/(b-I*a)*b/(a-I*b)*e^3*d^3*ln(1-(a+I*b)*exp(2*I
*(f*x+e))/(a-I*b))-6/f^2/(b-I*a)*b/(a-I*b)*e^2*c*d^2*x+I/f/(b-I*a)*b/(a-I*b)*d^3*ln(1-(a+I*b)*exp(2*I*(f*x+e))
/(a-I*b))*x^3+3/f^2/(b-I*a)*b/(a-I*b)*d^2*c*polylog(2,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x-2*I/f^4/(b-I*a)*b*e^
3*d^3/(I*b-a)*ln(exp(I*(f*x+e)))+3/2*I/f^3/(b-I*a)*b/(a-I*b)*d^2*c*polylog(3,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))
+3/2*I/f^3/(b-I*a)*b/(a-I*b)*d^3*polylog(3,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x+3/f^2/(b-I*a)*b/(a-I*b)*c^2*d*e
^2+3/2/f^2/(b-I*a)*b/(a-I*b)*c^2*d*polylog(2,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))+2/f^3/(b-I*a)*b/(a-I*b)*d^3*e^3
*x-4/f^3/(b-I*a)*b/(a-I*b)*e^3*c*d^2+3/2/f^2/(b-I*a)*b/(a-I*b)*d^3*polylog(2,(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))
*x^2+2*I/f/(b-I*a)*b*c^3/(I*b-a)*ln(exp(I*(f*x+e)))-I/f/(b-I*a)*b*c^3/(I*b-a)*ln(exp(2*I*(f*x+e))*a+I*b*exp(2*
I*(f*x+e))-a+I*b)+6*I/f^3/(b-I*a)*b*e^2*c*d^2/(I*b-a)*ln(exp(I*(f*x+e)))-3*I/f^3/(b-I*a)*b*e^2*c*d^2/(I*b-a)*l
n(exp(2*I*(f*x+e))*a+I*b*exp(2*I*(f*x+e))-a+I*b)-6*I/f^2/(b-I*a)*b*e*c^2*d/(I*b-a)*ln(exp(I*(f*x+e)))+3*I/f^2/
(b-I*a)*b*e*c^2*d/(I*b-a)*ln(exp(2*I*(f*x+e))*a+I*b*exp(2*I*(f*x+e))-a+I*b)+3*I/f/(b-I*a)*b/(a-I*b)*c^2*d*ln(1
-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x+3*I/f^2/(b-I*a)*b/(a-I*b)*c^2*d*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*e-
3*I/f^3/(b-I*a)*b/(a-I*b)*e^2*c*d^2*ln(1-(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))+3*I/f/(b-I*a)*b/(a-I*b)*d^2*c*ln(1-
(a+I*b)*exp(2*I*(f*x+e))/(a-I*b))*x^2+1/2/(b-I*a)*b/(a-I*b)*d^3*x^4+3/(b-I*a)*b/(a-I*b)*c^2*d*x^2+2/(b-I*a)*b/
(a-I*b)*d^2*c*x^3+3/2/f^4/(b-I*a)*b/(a-I*b)*d^3*e^4-3/4/f^4/(b-I*a)*b/(a-I*b)*d^3*polylog(4,(a+I*b)*exp(2*I*(f
*x+e))/(a-I*b))+d^2/(a+I*b)*c*x^3+3/2*d/(a+I*b)*c^2*x^2

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (203) = 406\).

Time = 0.34 (sec) , antiderivative size = 1031, normalized size of antiderivative = 4.26 \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3/(a+b*cot(f*x+e)),x, algorithm="fricas")

[Out]

1/8*(2*a*d^3*f^4*x^4 + 8*a*c*d^2*f^4*x^3 + 12*a*c^2*d*f^4*x^2 + 8*a*c^3*f^4*x - 3*I*b*d^3*polylog(4, ((a^2 + 2
*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^2 - 2*a*b - I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2)) + 3*I*b*d^3*polylog(4,
((a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (-I*a^2 - 2*a*b + I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2)) - 6*(-I*b*d
^3*f^2*x^2 - 2*I*b*c*d^2*f^2*x - I*b*c^2*d*f^2)*dilog(-(a^2 + b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (
-I*a^2 + 2*a*b + I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2) + 1) - 6*(I*b*d^3*f^2*x^2 + 2*I*b*c*d^2*f^2*x + I*b*c^2*
d*f^2)*dilog(-(a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^2 + 2*a*b - I*b^2)*sin(2*f*x + 2*e))/
(a^2 + b^2) + 1) + 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(1/2*a^2 + I*a*b - 1/2*b^2
- 1/2*(a^2 + b^2)*cos(2*f*x + 2*e) + 1/2*(I*a^2 + I*b^2)*sin(2*f*x + 2*e)) + 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f +
3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(-1/2*a^2 + I*a*b + 1/2*b^2 + 1/2*(a^2 + b^2)*cos(2*f*x + 2*e) + 1/2*(I*a^2 +
I*b^2)*sin(2*f*x + 2*e)) - 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2
*f + 3*b*c^2*d*e*f^2)*log((a^2 + b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (-I*a^2 + 2*a*b + I*b^2)*sin(2
*f*x + 2*e))/(a^2 + b^2)) - 4*(b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2
*f + 3*b*c^2*d*e*f^2)*log((a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a^2 + 2*a*b - I*b^2)*sin(2*
f*x + 2*e))/(a^2 + b^2)) - 6*(b*d^3*f*x + b*c*d^2*f)*polylog(3, ((a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2*e) + (I*a
^2 - 2*a*b - I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2)) - 6*(b*d^3*f*x + b*c*d^2*f)*polylog(3, ((a^2 - 2*I*a*b - b^
2)*cos(2*f*x + 2*e) + (-I*a^2 - 2*a*b + I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2)))/((a^2 + b^2)*f^4)

Sympy [F]

\[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\int \frac {\left (c + d x\right )^{3}}{a + b \cot {\left (e + f x \right )}}\, dx \]

[In]

integrate((d*x+c)**3/(a+b*cot(f*x+e)),x)

[Out]

Integral((c + d*x)**3/(a + b*cot(e + f*x)), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (203) = 406\).

Time = 0.57 (sec) , antiderivative size = 990, normalized size of antiderivative = 4.09 \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3/(a+b*cot(f*x+e)),x, algorithm="maxima")

[Out]

-1/12*(18*c^2*d*e*(2*(f*x + e)*a/((a^2 + b^2)*f) - 2*b*log(a*tan(f*x + e) + b)/((a^2 + b^2)*f) + b*log(tan(f*x
+ e)^2 + 1)/((a^2 + b^2)*f)) - 6*(2*(f*x + e)*a/(a^2 + b^2) - 2*b*log(a*tan(f*x + e) + b)/(a^2 + b^2) + b*log
(tan(f*x + e)^2 + 1)/(a^2 + b^2))*c^3 - (3*(f*x + e)^4*(a + I*b)*d^3 - 12*I*b*d^3*polylog(4, (I*a - b)*e^(2*I*
f*x + 2*I*e)/(I*a + b)) - 12*((a + I*b)*d^3*e - (a + I*b)*c*d^2*f)*(f*x + e)^3 + 18*((a + I*b)*d^3*e^2 - 2*(a
+ I*b)*c*d^2*e*f + (a + I*b)*c^2*d*f^2)*(f*x + e)^2 - 12*((a + I*b)*d^3*e^3 - 3*(a + I*b)*c*d^2*e^2*f)*(f*x +
e) + 12*(I*b*d^3*e^3 - 3*I*b*c*d^2*e^2*f)*arctan2(b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, a*cos(2*f*x + 2
*e) - b*sin(2*f*x + 2*e) - a) + 4*(-4*I*(f*x + e)^3*b*d^3 + 9*(I*b*d^3*e - I*b*c*d^2*f)*(f*x + e)^2 + 9*(-I*b*
d^3*e^2 + 2*I*b*c*d^2*e*f - I*b*c^2*d*f^2)*(f*x + e))*arctan2(-(2*a*b*cos(2*f*x + 2*e) + (a^2 - b^2)*sin(2*f*x
+ 2*e))/(a^2 + b^2), (2*a*b*sin(2*f*x + 2*e) + a^2 + b^2 - (a^2 - b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) + 6*(4*
I*(f*x + e)^2*b*d^3 + 3*I*b*d^3*e^2 - 6*I*b*c*d^2*e*f + 3*I*b*c^2*d*f^2 + 6*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x +
e))*dilog((I*a - b)*e^(2*I*f*x + 2*I*e)/(I*a + b)) + 6*(b*d^3*e^3 - 3*b*c*d^2*e^2*f)*log((a^2 + b^2)*cos(2*f*x
+ 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 + a^2 + b^2 - 2*(a^2 - b^2)*cos(2*f*x + 2*
e)) - 2*(4*(f*x + e)^3*b*d^3 - 9*(b*d^3*e - b*c*d^2*f)*(f*x + e)^2 + 9*(b*d^3*e^2 - 2*b*c*d^2*e*f + b*c^2*d*f^
2)*(f*x + e))*log(((a^2 + b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 +
a^2 + b^2 - 2*(a^2 - b^2)*cos(2*f*x + 2*e))/(a^2 + b^2)) - 6*(4*(f*x + e)*b*d^3 - 3*b*d^3*e + 3*b*c*d^2*f)*pol
ylog(3, (I*a - b)*e^(2*I*f*x + 2*I*e)/(I*a + b)))/((a^2 + b^2)*f^3))/f

Giac [F]

\[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \cot \left (f x + e\right ) + a} \,d x } \]

[In]

integrate((d*x+c)^3/(a+b*cot(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*x + c)^3/(b*cot(f*x + e) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{a+b \cot (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {cot}\left (e+f\,x\right )} \,d x \]

[In]

int((c + d*x)^3/(a + b*cot(e + f*x)),x)

[Out]

int((c + d*x)^3/(a + b*cot(e + f*x)), x)